Billy Collins has reported that he was influenced by the "contemporary" poets he was able to read in Poetry magazine growing up, people like Karl Shapiro, Howard Nemerov and Reed Whittemore. Along with them, he credits the movement and spirit of the cultural group the "Beats", or Beat Generation, from the late 1950's and early 1960's. This generation of people and writers valued spontaneity and open emotion, as well as a struggle against social conformity. They fought against organized religion and organized schooling, and are often referred to as the inspiration for the "hippies".
Besides those modern influences, Collins also reports that he wanted to be a newer Wallace Stevens, one of the first modernist poets. Stevens emphasized consciousness in his poetry, and wrote with little attention to conformed styles. Instead, he used obscure allusions and and attention to emotion.